Circuit complexity near critical points

TL;DR

This paper investigates the behavior of quantum circuit complexity near critical points in the Bose-Hubbard model, revealing peaks at phase transitions and scaling laws that connect complexity with critical exponents, supported by numerical and holographic methods.

Contribution

It introduces a method to compute circuit complexity in many-body systems near criticality, demonstrating scaling behavior and extending analysis beyond Gaussian approximations using holography.

Findings

Complexity peaks at critical points.

Scaling behavior of complexity near phase transitions.

Agreement with free field theory and holographic predictions.

Abstract

We consider the Bose-Hubbard model in two and three spatial dimensions and numerically compute the quantum circuit complexity of the ground state in the Mott insulator and superfluid phases using a mean field approximation with additional quadratic fluctuations. After mapping to a qubit system, the result is given by the complexity associated with a Bogoliubov transformation applied to the reference state taken to be the mean field ground state. In particular, the complexity has peaks at the $O(2)$ critical points where the system can be described by a relativistic quantum field theory. Given that we use a gaussian approximation, near criticality the numerical results agree with a free field theory calculation. To go beyond the gaussian approximation we use general scaling arguments that imply that, as we approach the critical point $t\rightarrow t_c$, there is a non-analytic behavior in the complexity $c_2(t)$ of the form $|c_2(t) - c_2(t_c)| \sim |t-t_c|^{\nu d}$, up to possible logarithmic corrections. Here $d$ is the number of spatial dimensions and $\nu$ is the usual critical exponent for the correlation length $\xi\sim|t-t_c|^{-\nu}$. As a check, for $d=2$ this agrees with the numerical computation if we use the gaussian critical exponent $\nu=\frac{1}{2}$. Finally, using AdS/CFT methods, we study higher dimensional examples and confirm this scaling argument with non-gaussian exponent $\nu$ for strongly interacting theories that have a gravity dual.